The sister could be bullying Naruto out of envy. If Naruto has some special abilities or talents that she doesn't, she might be jealous and lash out at him.
One possible reason could be to protect her. Naruto might believe that by leaving, he can draw danger away from his sister. For example, if there are powerful enemies targeting him, he doesn't want his sister to get caught in the crossfire.
One way it could be developed is that his sister might be jealous of Naruto's growing popularity or skills. Maybe she feels overshadowed and starts to bully him by spreading false rumors about him in the village. This could lead to Naruto feeling more isolated and misunderstood, even as he tries to prove himself.
One possible reason could be to protect her. Maybe she has some special power that attracts dangerous enemies, and Naruto, being the strong and protective character he is, seals her power to keep her safe from those who might misuse it.
Another possibility is that they leave because of a sense of disillusionment. They could have witnessed some corruption or unfairness within Konoha's leadership or among the ninja community. This makes them lose faith in the village and decide to start anew elsewhere.
It could be due to a misunderstanding. Perhaps there was an incident where his sister was involved, but Naruto misinterpreted her actions as something malicious against him. Like she might have accidentally broken something precious of his and he thought it was on purpose.
It could also be his boisterous personality. Naruto is very outgoing and sometimes a bit of a troublemaker in his attempts to get attention. This might annoy some of the more reserved or studious kids at school, leading them to bully him as a way to try and make him conform or be quiet.
She could be an original character created specifically for this fanfiction. The author might give her unique abilities and a distinct personality to make the story more interesting.
One possible reason could be jealousy. Maybe the sister is more favored by the villagers or has some special abilities that Naruto feels overshadow his own. This could lead to a sense of envy and ultimately hatred.
Maybe his sister is always favored by their parents, making Naruto feel left out and thus develop hatred.
It could be that his sister has some special abilities or talents that the family or the village values more. For example, if she has a very rare and powerful bloodline limit, everyone's attention would be drawn to her. In a ninja world, power and potential are highly regarded, so Naruto might be overlooked because his sister seems to have more of that 'special something' at first glance.